(Dear Antti: I understand it. hehe, in fact. for some unknown reason I can't sleep more than 5:40hrs excepting on Sundays.)

Dear Editor staff. Let me continue but now talking informally.

Take your time. It is not so trivial as it might look.

What this sequence is, and what it is not and should not be before proposing any further STRUCTURAL changes. I must emphasize always friendly and  humble of course, that I don't want to repeat the requiem lived with A215940 due the lot of meaningful, relevant, interesting, nice and valuable/noteworthy connections of such entry with another sequences and my particular inability to express in time what I was looking for (but finally and fortunately overcame)..... I proposed here only: ["Enumerations of 1,2,3,4,5,6...k in ascending order, excluding just one of them"] such table read by rows (indeed a triangle). These k permutations are a very special subset relative to the whole k! set of them when considered in lexicographic order due its structural implication in the outcomes obtained from the execution of the Narayana's algorithm. 

...well, with respect to the entry, It's clearly necessary an improvement to the picture. The picture is intended to be a visual complement for the example.

Hmm, I guess that some little check marks will be helpful there in order to throw the idea of counting the cells or elements in these matrices altogether.

At purpose, it is not the the main purpose to enumerate all the permutations for the first S naturals.

No, actually only those such that when all the S! permutations in lexicographic order are viewed such list, matrix, graph... (what applies) looks divided uniformly in S blocks.

Then the parallel alternative to computation arises naturally by the fact that we will know always where to start from and where to stop from and the exact number of times that the Narayana's algorithm should be executed by each thread/processor.

Well... I am not a master of the computer sciences neither got formal instruction in such field but I know a little about the efforts made in the search of parallel algorithms... also in this same thinking I guess that the entry should be kept as it was proposed, avoiding to mix it up too much with another already know sequences on enumerative combinatorics. Anyway the proper crossrefs should suffice.

I guess that If I gently write using LaTEX/PDF the particular calculation from where I observed the need of thinking the definition or name for this entry, then it will become clear. The big problem with explanations (imho) is to make people to FEEL the need of introducing/defining things.

Human beings (all of us fortunately), doesn't escape from the least action principle (in the sense stated by Maupertuis). If something looks not enough necessary, it will be disliked, refused, ignored and forgot. It is part of our evolutive nature.

And the simplicity should be the prime rule/directive.

Then my suggestion there is let us crossref interesting things instead of extending too much the entry.

About what the proposal is:

...If you have k naturals and exclude 1 of them, the somethings happen:

...k=1. You exclude the unique element. Then write "1" at first meaning this special place in the matrix that is the number to be excluded. Since there are no more elements the matrix ends as 1*1, only with such "1", conversely there is possible and logically correct the another interpretation I added below your edit in the example;

... Case k=2, this time you will have by excluding 1, the first and unique remaining element will be 2. So your first row will be "12". By excluding 2 instead, your first and unique element will be 1, then the second row will be "21", resulting matrix [[1,2],[2,1]]

... Case k=3. Since (k-1)=2 there will be two ranks "what comes at first" and "what comes at second". I called these numbers "j", "dot j", and "double dot j", with the trivial property that for any j, their product is 3! and their sum 3*(3+1)/2 or 6, and the special noteworthy  fact that there is the unique k for which this happens (that the sum of the first k naturals is the identical to k!).... Go on with the matrix: By excluding 1 the first and second remaining naturals between 1 and 3 in ascending order are 2 and 3 so according to our convention the first row in the matrix is "123". By applying the same when 2 is excluded instead of 1, then the second row is "213".... finally if 3 is excluded instead of either 1 or 2, the third row must be "312" Resulting in: [[1,2,3],[2,1,3],[3,1,2]]......

... Case k=4, There will be 4 rows:

By excluding: / There will be remaining in ascending order:
   "1"...................."234"
   "2"...................."134"
   "3"...................."124"
   "4"...................."123"
   
Resulting in: [[1,2,3,4],[2,1,3,4],[3,1,2,4],[4,1,2,3]]

Cool!. And why a(22) must be "4":

Answer: Build the matrices from 1 until the sum of all for all the elements for all these matrices be greater than 22.

[1]
[[1,2],[2,1]]
[[1,2,3],[2,1,3],[3,1,2]]
[[1,2,3,4],[2,1,3,4],[3,1,2,4],[4,1,2,3]]

Join the rows reading them from left to right and up to down as it was clarified by Mr. Karttunen:

[1][[1,2],[2,1]][[1,2,3],[2,1,3],[3,1,2]][[1,2,3,4],[2,1,3,4],[3,1,2,4],[4,1,2,3]] or simply

112211232133121234213431244123, then count 22 places from left to right, and look what is written there:

.112211232133121234213431244123;
.**********************--------;
   
There in PARI a function might be implemented doing exactly the same, preferably outside the constructor of a matrix object, but believe me once you got clear of this you will be agree that the recursive construction is the more easy way of generating this. I have realized of two iterative ways and the recursive illustrated (tried to be illustrated) with the picture.

I already took more than a year for the contemplation of this and other similar matters every night.

And I'm bringing only the most simple thoughts I was able to achieve in such period.
I finally decide to disclose publicly this after an unsuccessful search for reliable references like the Mr Knuth ones.

Feeling that I'm just in front of something new would be wrong.

I only did catch the property and exploit its capabilities as amateur programmer.

I'm not enough good (and well trained) to develop a Desktop application with GUI, but love the programming and the CS anyway, as much as the Mathematics itself. quoting Feynman once more <<just for..."the pleasure of finding things out">>.

Thanks for allowing me to share such passion.